I am a beginner in representation theory and algebraic geometry, so that references giving clear explanations of things like the tautological line bundle on $\mathbb P^n$, its dual, and the associated sheaves of sections $\mathcal O(-1)$ and $\mathcal O(1)$ would be especially welcome.

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The friendliest introduction to complex vector bundles on complex manifolds might be Complex Manifolds, Vector Bundles and Hodge Theory by Brylinski and Toth.

On the subject which interests you (link with representation theory), I recommend the article The Borel-Weil theorem for complex projective space by M.Eastwood and J.Sawon in the book Invitations to Geometry and Topology (Oxford Graduate Texts In Mathematics #7). The article is self-contained and fairly elementary.

You could also look at the nice book by Joseph Taylor, Several Complex Variables, published by the AMS, Graduate Studies In Mathematics #46. Borel-Weil is carefully handled there, with all the necessary preliminaries.

A brief introduction to line bundles (and divisors) on schemes is available at http://people.fas.harvard.edu/~amathew/linebund.pdf, although it is very rough. This covers things like the tautological line bundle on $\mathbb{P}^n$ and the sheaves $\mathcal{O}(n)$. In particular, it gives the correspondence (as in Hartshorne II.6) between Weil and Cartier divisors on regular noetherian integral schemes (such as projective space over a field). I don't know if this is completely relevant to what you need, but it may (or may not) be useful for your initial questions. Having not read Fulton's book, I can't answer your specific question.